Using the Screw modifier, how do I accurately set the axis of rotation?
The modifier uses the objects (local) axis (X, Y, Z)

Modeling from top side or front ortho view will make things simple. use numpad5 to switch orthogonal/perspective view

Shape modeled in top ortho view (green line is Y global axis)



Screw modifier set on Z object axis



Screw modifier set on Y object axis



Once modeled, you can move your object or rotate it (object mode) but in edit mode, you'll get a mess. Apply modifier or clear rotations for editing.